**TATJANA OSTROGORSKI**

**Slobodanka Jankovi´**

**c**

20 February 1950 – 25 August 2005

Tatjana Ostrogorski was born in Belgrade, in a family of distinguished
intel-lectuals. Both of her parents were historians, university professors and members
of the Serbian Academy of Sciences and Arts. Her mother Fanula Papazoglu came
from a Greek family from Bitolj, in Macedonia. Her father Georgije Ostrogorski, a
famous byzantologist and author of the book*History of the Byzantine State*, was
born in St. Petersburg. He was one of the most prominent Russian intellectuals
who found a home in Serbia after the Russian Revolution. Tatjana was married
to the logician Kosta Doˇsen, also a member of the Mathematical Institute. Their
daughter Ana is now 14 years old. Tatjana is also survived by a brother and a
sister.

Tatjana went to primary and secondary school of a classical orientation (with ancient Greek and Latin and not much mathematics). From her father, Tatjana learned Russian as her ﬁrst language. Her mother and her grandmother taught her modern Greek. She was also ﬂuent in French and English, and of course Ser-bian. In 1973, she graduated from the University of Belgrade at the Department

2000*Mathematics Subject Classiﬁcation.* 01A70; 26A12.
Supported by the Ministry of Science of Serbia, Grant 144021.

of Mathematics. She obtained her Master’s degree in 1978 with a thesis entitled

*Saturation Problems in the Theory of Approximation of Real Functions*. Tatjana
spent the academic year 1980/81 in Moscow at Moscow State University, where she
worked under the supervision of Sergei Borisovich Stechkin. In 1987, she obtained
her doctoral degree, with the thesis*Lp* *Inequalities with Weights on Cones in* *Rn*
*and* *Hp* *Spaces on Half-Planes inRn*. For both theses her advisor was Slobodan
Aljanˇci´c.

After graduating in 1973, Tatjana took up a position in Belgrade at the Math-ematical Institute, where she remained until the beginning of her long illness in 2003.

At the beginning of her career she also had teaching assistantships in diﬀer-ential geometry at the Department of Mathematics and in analysis at the Faculty of Mechanical Engineering of the University of Belgrade. In the academic year 1988/89, she taught a graduate course on Lie groups at the Faculty of Mathe-matics in Belgrade. From 1994 until 1998 Tatjana was a visiting professor at the University of Montpellier III, in France.

In 1982, Tatjana was appointed as Secretary (in eﬀect, Managing Editor) of

*Publications de l’Institut Math´ematique*. This appointment was based on
excep-tional qualities she had for this position: a patient and hard worker, with an
ex-cellent mathematical and general education, reliable, and an extremely decent and
kind person. She and Dragan Blagojevi´c, who was appointed at the same time as
Technical Editor of the journal, made a very good team for many years. Due to
their eﬀorts, the level and the appearance of the journal improved signiﬁcantly.

Tatjana was the Editor of the special issue of Publications de l’Institut Math´ e-matique, published in 2002, celebrating the centenary of Jovan Karamata’s birth. She was also a member of the Editorial Board for the publication of the works of Karamata, planned for 2002 (but still unpublished).

Tatjana’s research was in mathematical analysis. Her ﬁrst papers dealt with
Fourier analysis and approximation theory, while later ones are in regular
vari-ation, harmonic analysis and Lie groups. The results from her Master’s thesis
were published in [1], where she investigated saturation of convolution operators
(singular integrals) with kernels of the form *k _{r}*(

*x*) =

*rk*(

*rx*),

*r*

*∈*

*R*,

*k*

*∈*

*L*(

*R*), and she proved local and global saturation theorems for the spaces of temperate distributions (which are larger than

*Lp*spaces).

In her doctoral thesis, Tatjana investigated Hardy’s inequality for positive
func-tions on *R*:
+∞
0
*x−1*
*x*
0
*f*(*t*)*dt*
*p*
*xadxC*
+∞
0
*fp*(*x*)*xadx*

(*ap−*1,*C*- a certain const.), which says that the operator*Hf*(*x*) =*x−1*_{0}*xf*(*t*)*dt*

is continuous in the space*Lp*(0*,∞*), 1*p <∞*, with the weight*xa*. Her aim there
was not only to ﬁnd an *n*-dimensional analogue of Hardy’s inequality, but more
generally, to determine a class of operators as wide as possible, and weights such
that an inequality, analogous to Hardy’s, holds. She considered integral operators

deﬁned by

*Kf*(*x*) =

*V* *k*(*x, y*)*f*(*y*)*dy,* *x∈R*
*n*

where *k*: *V* *→R*+ is the kernel of the operator and*f* is a positive function on a
self-adjoint cone*V* in*Rn*. Her central result is that if*K*is a homogeneous operator
of order *b*, and for some*c*

*V* *k*
*r*_{(}* _{x}*
0

*, y*)

*s−rc*+

*r−1*(

*y*)

*dy <∞*holds, then

*Kf*

_{q,c}_{−}_{b}_{−1}Cf_{p,c},where the norm is *f _{p,µ}* = (

_{V}*|f*(

*x*)

*|psµp−1*(

*x*)

*dx*)1

*/p,*and 1

*p*

*q*

*∞*,

*r−1* =*p−1*+*q−1*, *C*- a certain const. This was applied to several*n*-dimensional
generalizations of classical operators such as those of Hardy, Laplace, Riemann–
Liouville, Hilbert and Stieltjes. The obtained inequalities, expressing the
continu-ity of the related operators in some *Lp* spaces with weights, reduce to the usual
inequalities when*n*= 1. These results were published in [7] and [8]. For weighted

*Hp*spaces, which are*n*-dimensional generalizations of Hardy spaces on half-planes,
she proved a theorem of the Paley–Wiener type in [9].

In the late 1970s, Slobodan Aljanˇci´c, Duˇsan Adamovi´c and Dragoljub
Aran-djelovi´c organized a special course on regular variation at the Department of
Math-ematics in Belgrade. Since then, Tatjana’s research was often connected with that
topic. Her ﬁrst papers in which regular variation appeared were [2] and [3], where
she investigated the integrability of the sum*g* of the sine (and also of the cosine)
series, depending on its coeﬃcients. In the case of quasimonotone coeﬃcients,
she presented in [2] suﬃcient conditions on the upper and lower indices of an

*O*-regularly varying function *K* such that *K*(1*/x*)*gp*(*x*) *∈* *L*(0*, π*) if and only if

*np−2K*(*n*)*ap _{n}*

*<∞*, where 1

*< p <∞*. An analogous result was obtained for the cosine series too. In the case of positive coeﬃcients, a similar theorem (with

*p*= 1) was obtained in [3].

Multidimensional generalizations of regularly varying functions occupy the
cen-tral part of Tatjana’s investigation. The ﬁrst results in this direction are Theorems
of the Abelian and Tauberian type for diﬀerent integral transforms of
multivari-ate regularly varying functions in the sense of Yakymiv1, published in [4], [5] and
[6]. Bajˇsanski and Karamata2deﬁned regularly varying functions on an arbitrary
topological group. In [11], Tatjana studied a special case when this topological
group is the group of automorphisms of a homogeneous cone *V* in *Rn*. She also
considered there regular variation with respect to the additive group of the vector

1_{A. L. Yakymiv,}_{Multidimensional Tauberian theorems and their applications to the }

*Bellman-Harris branching processes*, Mat. USSR Sb. 43 (1982), 413–425. See also Chapter 1.1 of Yakymiv’s
book*Probabilistic applications of Tauberian theorems*, Modern Probability and Statistics, VSP,
Utrecht, 2005.

2_{B. Bajˇ}_{sanski and J. Karamata,}_{Regularly varying functions and the principle of }

space*Rn*and established the relationship between the two classes by an exponential
mapping of the Lie group of the cone. The uniform convergence theorem and the
representation theorem were proved in this setup. In [12] and [13] regularly varying
functions deﬁned by Bajˇsanski and Karamata were studied on the light cone and
on the cone *Rn*_{+}. This deﬁnition of regular variation on *Rn*_{+} is diﬀerent than that
of Yakymiv. In [18], she explained the relationship between the Banach–Steinhaus
theorem and the uniform convergence theorem for regularly varying functions, gave
a simple proof of the uniform convergence theorem for continuous functions and
demonstrated that this proof can be carried on to general topological groups and
to Banach spaces.

In [10], Tatjana presented some results concerning Vinberg’s3 theory of
ho-mogeneous cones and hoho-mogeneous integrals, which she used in [14] and [15] to
prove *n*-dimensional generalizations of the Abelian type theorems for the Laplace
transform and, amongst others, for the Riemann–Liouville operator and for the
Stieltjes transform. In [16], she proved an*n*-dimensional generalization of Hardy’s
inequality for homogeneous cones.

In [17] and [21] regularly bounded functions were introduced. Various
prop-erties, analogous to the properties of classical regularly varying functions and *O*
-regularly varying functions, are proved for them: a uniform boundedness theorem,
a representation theorem, Abelian-type theorems for integral operators, a
Hardy-type inequality with regularly bounded weights.

The article [19] grew out of a problem posed by E. Omey, which is related to
the fact that the class of slowly varying functions is not closed under subtraction.
The question was: “If the truncated variance of a random variable is slowly varying,
are its components in the positive and in the negative half line slowly varying too;
or, in other words, given a nondecreasing slowly varying function *L*, which is the
sum of two nondecreasing functions*L*=*F*+*G*, does it follow that the components

*F* and *G* are necessarily slowly varying too?” In [19] one ﬁnds a necessary and
suﬃcient condition for the components to be slowly varying. The class of functions
satisfying this condition turned out to be rather small. In order to enlarge this
class, analogous problems, but with various stronger monotonicity conditions, were
treated in [22], [24] and [25]. The case where the functions above are nonincreasing
was presented in [20].

With Tatjana’s untimely death, the Mathematical Institute in Belgrade has lost a precious member, an excellent mathematician and the dearest of colleagues.

**Bibliography of Tatjana Ostrogorski**

[1] *Global and local saturation theorems in some spaces of temperate functions*, Publ. Inst. Math.,
Nouv. S´er.**26(40)**(1979), 199–213.

[2] *On integrability of trigonometric series with quasimonotone coeﬃcients*, Publ. Inst. Math.,
Nouv. S´er.**28(42)**(1980), 135–144.

[3] *Integrability theorems for trigonometric series with positive coeﬃcients*, Publ. Inst. Math.,
Nouv. S´er.**30(44)**(1981), 131–139.

3_{E. B. Vinberg,}_{The theory of convex homogeneous cones}_{, Tr. Mosk. Mat. O.-va 12 (1963),}

[4] *Abelian type theorems for some integral operators inRn*, Publ. Inst. Math., Nouv. S´er.

**35(49)**(1984), 93–103.

[5] *Asymptotic behaviour of Fourier transforms inRn*, Publ. Inst. Math., Nouv. S´er.**35(49)**
(1984), 105–117.

[6] *Asimptotika preobrazovaniya Furie funkcii mnogih peremennykh*, in: *Teoriya priblizhenii*
*funkcii, Sbornik s mezhdunarodnoj konferencii v Kieve 1983*, Nauka, Moskva, 1987, pp. 329–
332.

[7] *Weighted norm inequalities for Hardy’s operator and some similar operators inRn*, in: A.
Haar Memorial Conference (Budapest 1985), Colloq. Math. Soc. Janos Bolyai **49**(1987),
687–694.

[8] *Analogues of Hardy’s inequality inRn*, Studia Math.**88**(1988), no 3, 209–219.

[9] *WeightedHp-spaces on homogeneous halfspaces*, J. Math. Anal. Appl.**141**:1 (1989), 94–108.
[10] *Homogeneous cones and homogeneous integrals*, in: W. Bray et al. (eds),*Fourier Analysis*,

Marcel Dekker, New York, 1994, pp. 375–399.

[11] *Regular variation on homogeneous cones*, Publ. Inst. Math., Nouv. S´er. (Slobodan Aljanˇci´c
Memorial Volume)**58(72)**(1995), 51–70.

[12] *Regular variation on the light cone*, Math. Moravica (Special Volume (Proceedings of the 5th
International Workshop in Analysis and its Applications)) (1997), 71–84.

[13] *Regular variation inRn*_{+}, Mathematica (Cluj)**39(62)**:2 (1997), 267–276.

[14] *The Laplace transform on homogeneous cones*, Integral Transforms Spec. Funct. **6**(1997),
245–256.

[15] *Homogeneous cones and Abelian theorems*, Int. J. Math. Math. Sci.**21**:4 (1998), 643–652.
[16] *Weighted norm inequalities and homogeneous cones*, Colloq. Math.**77**:2 (1998), 251–264.
[17] *Regularly bounded functions and Hardy’s inequality*, in: W. O. Bray et al. (eds.),*Analysis of*

*Divergence*, Birkhauser, 1999, pp. 309–326.

[18] *Regular Variation and the Banach-Steinhaus Theorem*, Nieuw Arch. Wiskd., IV Ser.**16**:1-2
(1998), 27–36.

[19] (with S. Jankovi´c)*Diﬀerences of slowly varying functions*, J. Math. Anal. Appl.**247**:2 (2000),
478–488.

[20] (with S. Jankovi´c)*Diﬀerences of decreasing slowly varying functions*, Mat. Vesnik**52**(2000),
113–118.

[21] *On regularly bounded functions*, Proc. of the 10th Congress of Yugoslav Mathematicians,
2001, pp. 235–238.

[22] (with S. Jankovi´c)*Convex additively slowly varying functions*, J. Math. Anal. Appl.**274**:1
(2002), 478–488.

[23] (with S. Jankovi´c)*Two Serbian Mathematicians*, Math. Intelligencer (2002), p. 80.
[24] (with S. Jankovi´c) *Decomposition of convex additvely slowly varying functions*, Integral

Transforms Spec. Funct.**14**:4 (2003), 301–306.

[25] (with S. Jankovi´c)*The property of good decomposition in Hardy ﬁeld*, Indian J. Pure Appl.
Math.**35(10)**(2004), 1179–1183.

[26] (with S. Jankovi´c)*Good decomposition in the class of convex functions of higher order*, Publ.
Inst. Math., Nouv. S´er.**80(94)**(2006), 157–169.

Unpublished manuscripts:

[27] *Analyse sur les cones homogenes*, Universite Paul Valery, Montpellier III, Avril, 1993,
preprint.

[28] *Regularly varying functions and multidimensional generalizations*, contribution for Collected
Works of Jovan Karamata, in Serbian.